Optimisation of a generalised M(k)/M/k queueing system

We study a generalised M(k)/M/k queueing system where the arrival rate depends on the number of servers. There are holding costs reflecting the waiting cost of customers and service costs relating to operation of the servers. The objective is to minimise the total system cost in a steady state. We analyse the cost structure and show that the system holding cost rapidly decreases with the number of servers under some weak conditions. Based on this, we present an upper bound of the optimal number of servers. Finally, numerical examples are given to illustrate the model and the results.